for any vector fields \(X,Y\) . The map \(K\)  is called almost para-complex structure, and
it produces a splitting of the tangent bundle as \(TM=L\oplus \tilde{L}\) , where \(L, \tilde{L}\)  are the \((\pm 1)\) -eigenbundles of \(K\) .
These eigenbundles are maximally isotropic with respect to \(\eta \) , i.e. they are \(d\) -dimensional and satisfy (REF ),
and this implies that \(\tilde{L}\cong L^{*}\)  (see Section  for details).
If \(L\)  and \(\tilde{L}\)  are involutive under the Lie bracket then \(K\)  is said to be integrable,
and there exist local coordinates \((x^{i},\tilde{x}_{i})\) , \(i=1,...,d\) , such that,
denoting \(\partial _{i}=\partial /\partial x^{i}\)  and \(\tilde{\partial }^{i}=\partial /\partial \tilde{x}_{i}\) ,
one has \(L={\rm span}(\partial _{i})\)  and \(\tilde{L}={\rm span}(\tilde{\partial }^{i})\) .
If \(\eta \)  is flat, then it can be expressed in these coordinates as \(\eta ={\rm d}x^{i}\odot {\rm d}\tilde{x}_{i}\) .
So there are two complementary foliations \(F,\tilde{F}\)  such that \(L=TF\)  and \(\tilde{L}=T\tilde{F}\) 
This notation means that if \(\Sigma \)  is a leaf in \(F\) , then \(T\Sigma =L|_{\Sigma }\) , etc.,
and \(\tilde{L}\cong L^{*}\)  implies that \(T\tilde{F}\cong T^{*}F\) , thus \(TM\) 
has actually the structure of a generalized tangent bundle: \(TM\cong TF\oplus T^{*}F\) .
Furthermore, Vaisman shows [1]} that there is a natural notion of D-bracket on \(TM\) , and that it reduces to the
Dorfman bracket for fields with dependence only on \(F\) , so in this way one recovers generalized geometry.
Finally, different T-dual spacetimes correspond to different choices of para-complex structures.
